To find the values of $x$ when $y = 0$ for the equation

$y = (8 - x) \ln{x}$

we need to solve the equation $(8 - x) \ln{x} = 0$.

Step 1: Break the equation into factors

$(8 - x) \ln{x} = 0$ This equation will be zero if either factor is zero. Thus, we have two cases to consider.

Case 1: $\ln{x} = 0$

The natural logarithm of $x$ is zero when $x = 1$, since $\ln{1} = 0$.

So, one solution is: $x = 1$

Case 2: $8 - x = 0$

Solve for $x$: $x = 8$

Step 2: Combine the solutions

The solutions are $x = 1$ and $x = 8$.

Final Answer:

The values of $x$ when $y = 0$ are: $x = 1 \quad \text{and} \quad x = 8$

Would you like a detailed explanation of how these steps were performed?

Here are some related questions to extend your understanding:

1. What happens to the function as $x$ approaches 0 from the right?
2. Can $y = (8 - x) \ln x$ have negative solutions?
3. What is the domain of the function $y = (8 - x) \ln x$?
4. What is the behavior of the function as $x \to \infty$?
5. How does the graph of $y = (8 - x) \ln x$ look?

Tip: For any logarithmic function, the domain is limited to positive values of $x$, since $\ln{x}$ is undefined for $x \leq 0$.